Mathematics 2P0

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Mathematics 2P0
MATH29641, MATH29631
2010-11
Lecture 5
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Week 6 is Reading Week for 2P0/2Q1.
There will be a joint 2P0/2Q1 Lecture on
Monday Nov 1
st
at 4pm in Renold D7!
There will be no 2Q1 Lecture and no
2P0/2Q1 Tutorial on Wednesday Nov 3
rd
!
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2. Vector Calculus
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Vector Calculus

To describe mathematically many
electrical or electro-dynamic phenomena
which are observed in experiments the
‘toolbox’ of
vector calculus
is extremely
useful.
2. Vector Calculus
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2. Vector Calculus
2.1. Example Maxwell’s Equations
(i)
Gauss' law
describes the relationship between an
electric field
and the generating
electric charges
:
The electric field tends to point away from positive
charges and towards negative charges. More
technically, it relates the
electric flux
through any
hypothetical
closed
"
Gaussian surface
" to the electric
charge within the surface.
How can this statement be phrased in mathematical terms?
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2. Vector Calculus
2.1. Example Maxwell’s Equations
(ii)
Gauss' law for magnetism
states that there are no
"magnetic charges" (also called
magnetic monopoles
),
analogous to electric charges. Instead the magnetic
field is generated by a configuration called a
dipole
,
which has no magnetic charge but resembles a
positive and negative charge inseparably bound
together. Equivalent technical statements are that the
total
magnetic flux
through any Gaussian surface is
zero, or that the magnetic field is a
solenoidal
vector field
.
How can this statement be phrased in mathematical terms?
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2.1. Example Maxwell’s Equations
(iii)
Faraday's law
describes how a changing
magnetic field
can create ("induce") an
electric field
.
This aspect of
electromagnetic induction
is the
operating principle behind many
electric generators
: A
bar magnet
is rotated to create a changing magnetic
field, which in turn generates an electric field in a
nearby wire.
How can this statement be phrased in mathematical terms?
2. Vector Calculus
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2. Vector Calculus
2.1. Example Maxwell’s Equations
(iv)
Ampère's
law with Maxwell's correction
states that
magnetic fields can be generated in two ways: by
electrical current
(this was the original "Ampère's law")
and by changing electric fields (this was "Maxwell's
correction").
How can this statement be phrased in mathematical terms?
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2. Vector Calculus
2.1. Example Maxwell’s Equations


Maxwell’s equations can be formulated very
conveniently using concepts of
vector calculus
.


Typically, they can be written either in a
differential
form
using vector differential operators, or in
integral
form
.


These differential and Integral forms are related by
famous
vector integral theorems
which will be
discussed here as well.
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2.2.Scalar and Vector Fields

A
scalar field
assigns to each point in a
region of a 1D, 2D or 3D domain a scalar
value which is associated with this point.

Often this scalar value describes some
physical quantity which (ideally) can be
measured at this point.

Even if such a measurement is not done in
practice, a fictitious measurement can be
imagined which would give us this value.
2.2. Scalar and Vector Fields
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2.2.1. Scalar Fields
Examples

for scalar fields are:

The temperature at each point in this
lecture theatre.

The charge density at each point inside a
cathode ray tube.

The pressure at each point in the
atmosphere above Manchester.

The total X-ray radiation exposure of each
point in the body during a routine CT.
2.2. Scalar and Vector Fields
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2.2.1. Scalar Fields
Example for a scalar field in 2D:
Each point of the plate is assigned exactly one real
number, which describes the temperature at this point.
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2.2.2. Vector fields

A
vector field
is a distribution of vectors over a region
of space such that a vector is associated with each
point of the region.

Also these vectors often describe physical quantities
which (ideally) can be measured (sometimes in a
fictitious experiment) at every point.

Each of these measurements can be thought of
having a magnitude and a direction which both are
characteristic for each point.
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Example: Coulomb Force
2
1
4
3
2
1
0
2
1
r
r
r
r
q
q
F


=
 
1
r
Coulomb's law
is a
law
of
physics
describing the
electrostatic interaction between
electrically charged
particles.
In order to obtain both the magnitude and direction of
the force on a charge,
q
1 at position , experiencing a
field due to the presence of another charge,
q
2 at
position , the full
vector
form of Coulomb's law is
2
r
2.2.2. Vector Fields
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2.2. Scalar and vector fields
1
q
2
q
A graphical representation of Coulomb's law. You
get a vector field if you keep fixed and move
around and verify the Coulomb force at each
location of .
2
q
2.2.2. Vector Fields
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A visualization of the Coulomb force field could look like this:
2.2.2. Vector Fields
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Other examples for vector fields are

The water velocity at each point in a river
2.2.2. Vector Fields
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The gravitational pull of the earth
2.2.2. Vector Fields
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If the value (vector) assigned to a given
point does not depend on the time when a
(fictitious) measurement is made, the field
is said to be
stationary
.

If the value (vector) changes with time,
then the field is said to be
time-dependent
.
2.2. Scalar and Vector Fields
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In the following we will study the properties
of scalar and vector fields, in particular
arising from
electromagnetic phenomena
.


However, the
mathematical concepts
used
in our description are
far more general
, and
can be used as well for describing different
physical phenomena such as
Fluid Flow
,
Elasticity
,
Traffic Control
,
Molecular
Dynamics
, …
2.2. Scalar and Vector Fields
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Further Reading:

HELM modules 28.1 and 28.2.